Inverse Function . Symbol f -1(x) means the. inverse of f(x)  Inverse Function  Symbol f -1(x) means the inverse of f(x)  f  f -1(x) = x  f -1  f(x) = x  Cancel each other out

IMPORTANT The notation f -1(x) means the inverse of a function IMPORTANT The notation f -1(x) means the inverse of a function. The -1 is not a power, and the notation does not mean reciprocal.

Common sense inverses …  + and –  X and   Square and square root

For instance … 𝑓 𝑥 =5𝑥−7 and 𝑔 𝑥 = 𝑥+7 5 are inverses.

𝑓 𝑥 =5𝑥−7. 𝑔 𝑥 = 𝑥+7 5 Find g  f(x). 𝑔 5𝑥−7 = 5𝑥−7+7 5 = 5𝑥 5 𝑓 𝑥 =5𝑥−7 𝑔 𝑥 = 𝑥+7 5 Find g  f(x) 𝑔 5𝑥−7 = 5𝑥−7+7 5 = 5𝑥 5 = x Similarly f  g(x) = x =𝑥

You can find the inverse of a function by switching around “x” and “y” You can find the inverse of a function by switching around “x” and “y”. So the inverse of y = x2 is x = y2. __ or +  x = y

Note that the invese of a function is not necessarily a function.

In order for the inverse to be a function, the original must be a one-to-one function (every y is paired with a unique x)

Is the inverse of y = | x| a function?

Is the inverse of y = | x| a function? No, because | x | = | -x |

Is the inverse of y = x3 a function?

Is the inverse of y = x3 a function Is the inverse of y = x3 a function? Yes, because every y is paired with a unique x

Note that the function and its inverse are reflections of each other through the line y = x (a 45o angle). This is always true.

Graph the inverse of 𝑦= 1 𝑥 2

Graph the inverse of 𝑦= 1 𝑥 2 Remember the original function looks like this:

Graph the inverse of 𝑦= 1 𝑥 2 We will reflect this through the line y = x

Graph the inverse of 𝑦= 1 𝑥 2 The reflection looks like this:

Graph the inverse of 𝑦= 1 𝑥 2 So the graph of the inverse is:

Find the inverse of { (3,7) , (2,-5) , (-7,1) , (0,0) }

Find the inverse of { (3,7) , (2,-5) , (-7,1) , (0,0) } { (7,3) , (-5,2) , (1,-7) , (0,0) } Just switch x’s and y’s around.

Find the inverse of 𝑦= 1 𝑥+3

Find the inverse of 𝑦= 1 𝑥+3 First switch around x and y 𝑥= 1 𝑦+3

Now solve for y 𝑥= 1 𝑦+3 𝑥(𝑦+3)=1 𝑦+3= 1 𝑥 𝑦= 1 𝑥 −3

The domain of the original function is the range of the inverse (and vice versa).

Assuming f(x) maps the green set. Find the domain to the Assuming f(x) maps the green set Find the domain to the and range of blue f(x) and f -1(x) set …

Domain of f(x) { 1, 2, 3, 4, 5 } Range of f(x) { 2, 4, 6, 7, 10 }

Domain of f -1(x). { 2, 4, 6, 8, 10 } Range of f -1(x)